Confinement energy ($\mathcal{V}[\mu ]=\int V(x)d\mu (x)$)

The average of a confinement potential against a probability measure. A potential bounded below with at most quadratic growth has finite energy on probability measures with finite second absolute moments. A lower Hessian matrix bound $D^{2}V\ge \lambda I$ gives $\lambda$-displacement convexity: apply convexity of $V-\lambda |\cdot {|}^2/2$ along each particle segment in an optimal transport plan, then integrate. Quadratic growth of $V$ alone does not justify two differentiations under the energy integral.